A,B,C,.... Some of these corresponding sets and probability meanings are
given in Table 2.1. As Table 2.1 shows, the empty set is considered an
impossible event since no possible outcome is an element of the empty set.
Also, by ‘occurrence of an event’ we mean that the observed outcome is an
element of that set. F or example, event is said to occur if and only if the
observed outcome is an element of or or both.
Example 2.4.Consider an experiment of counting the number of left-turning
cars at an intersection in a group of 100 cars. The possible outcomes (possible
numbers of left-turning cars) are 0,1,2,...,100. Then, the sample space S is
. Each element of S is a sample point or a possible out-
come. The subset is the event that there are 50 or fewer
cars turning left. The subset is the event that between 40
and 60 (inclusive) cars take left turns. The set is the event of 60 or fewer
cars turning left. The set is the event that the number of left-turning cars
is between 40 and 50 (inclusive). Let Events A and C are
mutually exclusive since they cannot occur simultaneously. Hence, disjoint sets
are mutually exclusive events in probability theory.
2.2.1 Axioms of Probability
We now introduce the notion of a probability function. Given a random experi-
ment, a finite number P(A) is assigned to every event A in the sample space S of
all possible events. The number P(A) is a function of set A and is assumed to
be defined for all sets in S. It is thus a set function, and P(A) is called the
probability measure of A or simply the probability of A. It is assumed to have the
following properties (axioms of probability):
Table 2.1 Corresponding statements in set theory and probabilitySet theory Probability theory
Basic Probability Concepts 13
Space,S Sample space, sure event
Empty set,; Impossible event
Elementsa,b,... Sample pointsa,b,... or simple events)
SetsA,B,... EventsA,B,...
A EventAoccurs
A EventAdoes not occur
A[B At least one ofAandBoccurs
AB BothAandBoccur
ABAis a subevent ofB i.e. the occurrence ofAnecessarily implies
the occurrence ofB)
AB; AandBare mutually exclusive i.e. they cannot occur
simultaneously)
;
A[B
A B
Sf0, 1, 2,..., 100
Af0, 1, 2,...,50
Bf40, 41,...,60
A[B
A\B
Cf80, 81,..., 100g
ggg